Formulate a linear model for the following problem. No need to solve it$,$ but points will only be awarded if the model is completely correct. Don't forget the non$-$negativity constraints. Hint: It is not necessary for all variables to have the same letter, nor to be a single letter.

In the next three months, a steel company must meet the following demands: $100$ tons for the first month, $200$ tons for the second, and $50$ tons for the third. In any given month, an employee can produce up to $15$ tons of steel. Each employee has a salary of $$5000$ per month. Employees can be hired or fired at any time $($assuming immediate hiring$)$ at a cost of $$3000$ per dismissal and $$4000$ per hiring.

The steel can stay in inventory for $$100$ per ton per month. Additionally, the demand can be met late at a cost of $$70$ for each month of delay. That is$,$ if the demand for the first month is not met until the third, there is a delay cost of $$140$ per ton.

At the start of the first month, the company has $8$ employees hired and can hire a maximum of $2$ employees per month. All demand must be met by the end of the third month. The raw material to produce one ton of steel costs $$300.$

Formulate a linear model to minimize costs.